Example 1

Draw a rectangular prism with the following points as the base: L (6, 0, 0), M (6, 0, 5), N (0, 0, 5), and O (0, 0, 0). The height of the prism is 8 units. Then find its volume.

Hint

The height will be the y-axis coordinate. The volume is the area of the base times the height.

Answer

The prism has a volume of 240 units3 and looks like this:

Example 2

The coordinates of a cube are S (0, 0, 0), T (0, 0, 4), U (4, 0, 4), V (4, 0, 0), W (0, 4, 0), X (0, 4, 4), Y (4, 4, 4), and Z (4, 4, 0). If the cube was reduced by a scale factor of 4 and translated 4 units up the y-axis, what would be the coordinates of the new cube?

Hint

For dilation or reduction, multiply all numbers by the scale factor (or its inverse). For translation into another language, a dictionary would be useful. For translating the figure, add or subtract however many units to the proper number in the coordinates.

Example 3

A right square pyramid has a base at points A (6, 0, 6) B (6, 0, -2), C (-2, 0, 6), and D (-2, 0, -2). If the slant height of the pyramid is 8.06 units, what are the coordinates of the vertex?

Hint

The vertex must be in the middle of the square base, which gives us two of its coordinates. The height depends on the slant height.

Answer

The vertex is at (2, 2, 7)

Example 4

Graph a rectangular prism that contains the origin (0, 0, 0) and the point G (5, 7, 3) as vertices. Then, find its volume.

Hint

The vertices of the rectangular prism will have to be (x, y, z), where each of the numbers are either the same as in G (5, 7, 3) or the same as in the origin (0, 0, 0).

Answer

The rectangular prism has a volume of 105 units3 and looks like this

Example 5

Clarisse and Lorraine, two chickens, get into a fight. They were right next to each other, but Clarisse walks 7 feet north, 3 feet east, and flutters onto a fence that's 2 feet tall. Lorraine walks 1 foot north (to try and make up with Clarisse), but then decides against it and walks 9 feet west. What's the smallest distance between them?

Hint

If the north-south-east-west is set to the x-y plane, and elevation is given to the z plane, we can find the points of Clarisse and Lorraine and then calculate the distance between them.